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 A294828 Numerators of the partial sums of the reciprocals of the numbers (k + 1)*(5*k + 3) = A147874(k+2), for k >= 0. 4
 1, 19, 263, 815, 95597, 678149, 7531399, 18016577, 259695727, 4173941423, 222039686299, 2153029760377, 19428099753313, 331021112488901, 24211723390477517, 12126560607807901, 1008024074147249303, 168229609596886043, 1740462375346732831, 1219642439745618215 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The corresponding denominators are given in A294829. For the general case V(m,r;n) = Sum_{k=0..n} 1/((k + 1)*(m*k + r)) = (1/(m - r))*Sum_{k=0..n} (m/(m*k + r) - 1/(k+1)), for r = 1, ..., m-1 and m = 2, 3, ..., and their limits see a comment in A294512. Here [m,r] = [5,3]. The limit of the series is V(5,3) = lim_{n -> oo} V(5,3;n) = ((5/2)*log(5) - (2*phi-1)*(log(phi) + (Pi/5)*sqrt(7-4*phi)))/4, with the golden section phi:= (1 + sqrt(5))/2 = A001622. The value is 0.48170177449... given in A294830. In the Koecher reference v_5(3) = (2/5)*V(5,3) = 0.19268070979833151082... given there by ((1/4)*log(5) - (1/(2*sqrt(5)))*log((1+sqrt(5))/2) - (Pi/10)*sqrt((5 - 2*sqrt(5))/5)). REFERENCES Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189 - 193. LINKS Robert Israel, Table of n, a(n) for n = 0..891 Eric Weisstein's World of Mathematics, Digamma Function FORMULA a(n) = numerator(V(5,3;n)) with V(5,3;n) = Sum_{k=0..n} 1/((k + 1)*(5*k + 3)) = Sum_{k=0..n} 1/A147874(k+2) = (1/2)*Sum_{k=0..n} (1/(k + 3/5) - 1/(k+1)) = (-Psi(3/5) + Psi(n+8/5) - (gamma + Psi(n+2)))/2 with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620. EXAMPLE The rationals V(5,3;n), n >= 0, begin: 1/3, 19/48, 263/624, 815/1872, 95597/215280, 678149/1506960, 7531399/16576560, 18016577/39369330, 259695727/564293730, 4173941423/9028699680, 222039686299/478521083040, 2153029760377/4625703802720, 19428099753313/41631334224480, ... V(5,3;10^6) = 0.4817015746 to be compared with 0.4817017745 from A294830 with 10 digits. MAPLE map(numer, ListTools:-PartialSums([seq(1/(k+1)/(5*k+3), k=0..50)])); # Robert Israel, Nov 17 2017 PROG (PARI) a(n) = numerator(sum(k=0, n, 1/((k + 1)*(5*k + 3)))); \\ Michel Marcus, Nov 17 2017 CROSSREFS Cf. A001620, A001622, A147874, A294512, A294826/A294827 (V(5,2;n)), A294829, A294830, Sequence in context: A081036 A244652 A142817 * A016257 A021104 A209075 Adjacent sequences: A294825 A294826 A294827 * A294829 A294830 A294831 KEYWORD nonn,frac,easy AUTHOR Wolfdieter Lang, Nov 16 2017 STATUS approved

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Last modified December 7 18:28 EST 2023. Contains 367660 sequences. (Running on oeis4.)